## The Annals of Statistics

### Asymptotic Behavior of $M$ Estimators of $p$ Regression Parameters when $p^2 / n$ is Large; II. Normal Approximation

Stephen Portnoy

#### Abstract

In a general linear model, $Y = X\beta + R$ with $Y$ and $R n$-dimensional, $X$ a $n \times p$ matrix, and $\beta p$-dimensional, let $\hat\beta$ be an $M$ estimator of $\beta$ satisfying $0 = \sum x_i\psi(y_i - x'_i\beta)$. Let $p \rightarrow \infty$ such that $(p \log n)^{3/2} /n \rightarrow 0$. Then $\max_i|x'_i(\hat{\beta} - \beta)| \rightarrow _P 0$, and it is possible to find a uniform normal approximation for the distribution of $\hat{\beta}$ under which arbitrary linear combinations $a'_n (\hat{\beta} - \beta)$ are asymptotically normal (when appropriately normalized) and $(\hat{\beta} - \beta)'(X'X)(\hat{\beta} - \beta)$ is approximately $\chi^2_p$.

#### Article information

Source
Ann. Statist., Volume 13, Number 4 (1985), 1403-1417.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176349744

Digital Object Identifier
doi:10.1214/aos/1176349744

Mathematical Reviews number (MathSciNet)
MR811499

Zentralblatt MATH identifier
0601.62026

JSTOR

Subjects
Primary: 62G35: Robustness
Secondary: 62E20: Asymptotic distribution theory 62J05: Linear regression

#### Citation

Portnoy, Stephen. Asymptotic Behavior of $M$ Estimators of $p$ Regression Parameters when $p^2 / n$ is Large; II. Normal Approximation. Ann. Statist. 13 (1985), no. 4, 1403--1417. doi:10.1214/aos/1176349744. https://projecteuclid.org/euclid.aos/1176349744

• Part I: Stephen Portnoy. Asymptotic Behavior of $M$-Estimators of $p$ Regression Parameters when $p^2/n$ is Large. I. Consistency. Ann. Statist., Volume 12, Number 4 (1984), 1298--1309.
• See Correction: Stephen Portnoy. Correction: Asymptotic Behavior of $M$ Estimators of $p$ Regression Parameters when $p^2 / n$ is Large: II. Normal Approximation. Ann. Statist., Volume 19, Number 4 (1991), 2282--2282.